3.2 Matter Excitations
71
To show the role of the effective mass (see Fig. 3.5, in analogy to the 3-axes version
in [81]), the equation for the critical temperature for BEC in an ideal 3D system with
massive bosons is given here:
k B T critical =
2π
2
m particle
n particle
2.612
2/3 ,
(3.5)
with k B the Boltzmann constant, n particle the particle density and m particle the particle
(effective) mass (see e.g. [131]).
Only due to a long stamina and repeated efforts were the experimental signatures obtained with credible results [138–142]. Recently, interlayer excitons in 2Dsemiconductors type-II heterostructures have been targeted as a new platform for
exciton condensation and superfluidity studies [143, 144] or 2D electronic systems
in quantum Hall bilayers [145–147].
On Quantum Emitters and Quantum Droplets
If strongly localised in three directions, excitons can act very nicely as quantum
emitters (single-photon emitters) due to the modified energy spectrum in quantum
boxes (0D structures, referred to as quantum dots) [12, 16, 19, 20]. Quantum dots
can be also found in ultra-short-pulse semiconductor laser chips and in optical elements for such systems [148–150], such as saturable absorbers, and even provide
record high output powers if delivered in multi-stacking configuration as heavilypumped active layers in semiconductor disk lasers [151–153]. In most cases, however, excitons are utilised in 2D systems such as quantum wells, which confine
the charge carriers only in one direction (see for instance [87, 154, 155]). Beyond
single correlated electron–hole pair entities in quantum-well systems, even higher
correlations—quantum droplets of electrons and holes, referred to as “dropletons”
in quantum systems—were studied in the literature in non-equilibrium conditions of
an excited electron–hole plasma [156].
Coherent Excitons and Exciton–Polaritons
Optical transitions and light propagation are directly related to the polarisability
and susceptibility of a medium. In classical and semiclassical linear optics, the concept of the harmonic oscillator with Lorentzian lineshape and oscillator strength
( f ∝ N |M|
2 , with N the number of contributing resonant dipole oscillators) provides a very effective approach to describe light–matter interactions and particularly
macroscopic polarisation waves propagating through matter (see e.g. [87] for a comprehensive description and discussion of semiconductor optics). By employing the
implicit polariton equation (also see [88, 157, 158])
c
2 k
2
ω 2 = b +
i
f i
ω
2
0,i (+2ω 0,i Ak 2 ) − ω 2 + iωγ i
,
(3.6)
based on ω = ck and k
2
= k
2
= ˜
n
2
(ω) · k
2
vacuum = (ω) · (ω/c)
2 (with complex
functions ˜
n
2
(ω) = (ω)), the light–matter states’ dispersion relation in the crystal can be derived without and with spatial dispersion. The latter is represented in
71
To show the role of the effective mass (see Fig. 3.5, in analogy to the 3-axes version
in [81]), the equation for the critical temperature for BEC in an ideal 3D system with
massive bosons is given here:
k B T critical =
2π
2
m particle
n particle
2.612
2/3 ,
(3.5)
with k B the Boltzmann constant, n particle the particle density and m particle the particle
(effective) mass (see e.g. [131]).
Only due to a long stamina and repeated efforts were the experimental signatures obtained with credible results [138–142]. Recently, interlayer excitons in 2Dsemiconductors type-II heterostructures have been targeted as a new platform for
exciton condensation and superfluidity studies [143, 144] or 2D electronic systems
in quantum Hall bilayers [145–147].
On Quantum Emitters and Quantum Droplets
If strongly localised in three directions, excitons can act very nicely as quantum
emitters (single-photon emitters) due to the modified energy spectrum in quantum
boxes (0D structures, referred to as quantum dots) [12, 16, 19, 20]. Quantum dots
can be also found in ultra-short-pulse semiconductor laser chips and in optical elements for such systems [148–150], such as saturable absorbers, and even provide
record high output powers if delivered in multi-stacking configuration as heavilypumped active layers in semiconductor disk lasers [151–153]. In most cases, however, excitons are utilised in 2D systems such as quantum wells, which confine
the charge carriers only in one direction (see for instance [87, 154, 155]). Beyond
single correlated electron–hole pair entities in quantum-well systems, even higher
correlations—quantum droplets of electrons and holes, referred to as “dropletons”
in quantum systems—were studied in the literature in non-equilibrium conditions of
an excited electron–hole plasma [156].
Coherent Excitons and Exciton–Polaritons
Optical transitions and light propagation are directly related to the polarisability
and susceptibility of a medium. In classical and semiclassical linear optics, the concept of the harmonic oscillator with Lorentzian lineshape and oscillator strength
( f ∝ N |M|
2 , with N the number of contributing resonant dipole oscillators) provides a very effective approach to describe light–matter interactions and particularly
macroscopic polarisation waves propagating through matter (see e.g. [87] for a comprehensive description and discussion of semiconductor optics). By employing the
implicit polariton equation (also see [88, 157, 158])
c
2 k
2
ω 2 = b +
i
f i
ω
2
0,i (+2ω 0,i Ak 2 ) − ω 2 + iωγ i
,
(3.6)
based on ω = ck and k
2
= k
2
= ˜
n
2
(ω) · k
2
vacuum = (ω) · (ω/c)
2 (with complex
functions ˜
n
2
(ω) = (ω)), the light–matter states’ dispersion relation in the crystal can be derived without and with spatial dispersion. The latter is represented in